生产者理论1-课件

李永波TheFirm:ProductionAdvancedMicroEconomicsInthislecturewesetoutsomeoftheelementsneededforananalysisofthefirmTechnicalefficiencyReturnstoscaleConvexitySubstitutabilityMarginalproducts...and(fornexttime)assumingacompetitiveenvironment.Wedoitwithinthecontextofasingle-outputfirm...Butfirstweneedthebuildingblocksofamodel...Thebasicsofproduction...ziamountofinputiQamountofoutputThebasicsof

production...inputvectorz:=(z1,z2,...,zm)wipriceofinputiw:=(w1,w2,...,wm)inputpricevectorPpriceofoutputNotation:PricesQ£G(z1,z2,....,zm)Thesingle-outputproductionfunctionWrittenmorecompactlyQ£

G(z)technologyoutputYes,butwhynot"="signhere?inputsThemeaningofthefunctionthemaximumamountofoutputthatcanbeproducedfromthislistofinputsUsethisrelationtodistinguishtwocases...ComponentsoftherelationshipQ<G(z)Q=G(z)21ThecasewhereproductionistechnicallyefficientThecasewhereproductionis(technically)inefficientTechnicalefficiency¶G(z)

¶zi____iG(z):=

wheredifferentiableSomehandynotation...z2Qz10

G(z,z)12outputinput2input1Q<G(z,z)12Feasible,butinefficientpointsQ>G(z,z)12infeasiblepointsQ=G(z,z)12technicallyefficientpointsNowlet’sslicethissetuptogetsomeusefultoolsThefullproductionfunctionPickaparticularoutputlevelQZ(Q):={z:G(z)³Q}Findafeasibleinputvectorz

G(z)

³QRepeattofindallsuchinputvectorsInputrequirementsetsZ(Q):={z:G(z)³Q}thesetofinputvectorsthatmeetthetechnicalfeasibilityconditionforoutputlevelQ......butwhatwouldZlooklike??Thisdependsontheassumptionswemakeaboutproduction...First,a“standard”case...?Whatisthisthing...?z1z2G(z,z)=`Q12

infeasiblepoints_Z(Q)technicallyefficientpoints

feasible,butinefficientpointsG(z,z)<`Q12G(z,z)>`Q12Theinputrequirementset:z1z2

z¢

z²

PicktwoboundarypointsDrawthelinebetweenthemIntermediatepointsmustlieintheinteriorofZmeaning:acombinationoftwotechniquesmayproducemoreoutputG(z¢)=`QG(z²)=`QG(z)>`QButwhatifwechangedsomeoftheassumptionshere?_Z(Q)Case1:Z

issmoothandstrictlyconvexz1z2_Z(Q)

z¢

z²

PickanytwopointsinZDrawthelinebetweenthemIntermediatepointsmustlieinZmeaning:acombinationoffeasibletechniquesisalsofeasibleCase2:Z

convexbut

notstrictlyconvexz1z2

thispointisnotfeasible_Z(Q)Thisregioncausesaproblemmeaning:inthisregionthereisanindivisibilityCase3:Z

issmoothbutnotconvexAnExample...LondonNewYork3131z1z2slopeundefinedatthispoint

theonlyefficientpointfor

Q=`Q_Z(Q)Case4:Z

isconvexbutnotsmoothz1z2z1z2z1z2z1z2Standardcase,butstrongassumptionsaboutdivisibilityandsmoothnessalmostconventionalcase:mixturesmaybejustasgoodassingletechniquesPresentsproblems:the"dent"representsanindivisibilityunusualcase:onlyoneefficientpointandnotsmooth.Butnotperverse.Summary:4possibilitiesfortheinputrequirementsetZ

{z:G(z)=Q}Thisistheisoquant.Let'slookatitsshape...PickanoutputlevelQ

FindtheinputrequirementsetZ(Q)

DrawtheboundaryofthissetIsoquants{z:G(z)=Q}Az1inputsrequiredtoproduceatAz2IsoquantatQSlope=z2/z1TheinputratiodescribestheparticulartechniqueTheisoquantistheboundaryofZTheisoquantthroughA

(Q)MarginalrateoftechnicalsubstitutionTheslopeoftheisoquantisthemarginalrateofsubstitutionatA.Itmeasurestheimplicit“price”ofinput1intermsofinput2.Thehigheristhis“price”,thesmalleristherelativeusageofinput1z1z2

A

A'G1(z)/G2(z)

TheresponsivenessoftheinputratiototheMRTSisgivenbytheelasticityofsubstitution-

log(z1/z2)

log(G1/G2)Canbeseenastheisoquant’s“curvature”inputratioNowforaspecialcase...Aconstantelasticityofsubstitution:Increasetheelasticityofsubstitution...z1z2Nowlookatthestructureofthecontourmap...z1z2Homotheticcontoursz1z2Qtz2tztQr1G(tz)=tG(z)rContoursofahomogeneousfunctionTheisoquantsformacontourmap.Ifwelookedatthe“parent”diagram,whatwouldwesee?Let'sdothisfor2inputs,oneoutput.Let'srebuildfromtheisoquantsz2Qz1expansionray

0G(tz)=tG(z)constantreturnstoscaleProportionalincreaseinallinputsz2Qz10t>1Þ

G(tz)>tG(z)IncreasingreturnstoscaleProportionalincreaseinallinputsz2Qz10t>1ÞG(tz)<tG(z)DecreasingreturnstoscaleProportionalincreaseinallinputsz2Qz1isoquantQ=`Q0Takea"horizontal"section...togettheisoquantagain

z2Qz10…thisgivesusournextconceptNowtakea"vertical"section...

Pickatechnicallyefficientinputvector

Keepallbutoneinputconstant

Q=G(z)Measurechangeinoutputw.r.t.thisinput¶G(z)¶zi____MPi=Gi(z)=Marginalproductsz1QG(z)z1QG(z)z1QG(z)possiblerelationshipsbetweenoutputandoneinputz1QG(z)Let'staketheconventionalcase…feasiblesetG(z)Qz1input1isessentialSetoftechnicallyefficientpointsTaketherelationshipbetweenoutputandinput1...Ifz1=0thenQ=0z1QG(z)

G1

fallswithz1

ifGisconcaveMarginalproductslope=G1(z)

Technicalefficiency

That'sfornexttime...Returnstoscale

Convexity

MRTSMarginalproductAlloftheseareimportantinthefirm'soptimisat